Along with the push buttong and bell, there's a transformer somewhere in the circuit, and this may be why your bell's not working. This is a small metal boxy thing, about 3" on a side. The transformer could be most anywhere, but some typical locations are in the garage, in a coat closet just near the front door or door bell, in the furnace closet, in the attic. (note: some doorbell chimes have the transfomer built into the chime itself, but I've yet to come across one like that personally - it's always been an easter egg hunt to find the remote transformer).

If you've got a multimeter, you can check for voltage on either the button or chime wires. It's DC voltage, about 12 volts (I've seen some transformers at 6 or 8 volts). If you Don't have a multimeter, an easy check for the transformer is the if the button lights up on your door bell button (assuming you've got a light-up button :). If the button doesn't light up, good chances the transformer's shot and that's why the bell wasn't working in the first place.

If it's the transfomer, replacement is pretty easy - just remove the wires from the old transformer and reconnect to the new in the same way. HOWEVER, there is a risk of shock! The transformer provides 12 volts to the button and chime, but it's powered by 110 volts. Some transformers plug into a receptacle, in which case you simple have to unplug the old before disconnect wires. But typically the transformer is hard-wired into a junction box. If this is the case, you'll need to turn off the breaker/fuse that powers it. I'd suggest picking up a circuit tester when you get the transfomer. It's a plastic probe about the size of a Sharpie pen, and lights up and makes a tone when held near a "hot" wire. That way you can tell for sure that you've cut the power to the junction box before you start poking around.